Complete Bitcoin noob here. What are the advantages of having a large network of miners on the Bitcoin network? Why not have a limited number of super-peers that comprise the network? Isn't that the direction that things are heading anyway, given that only a select few will have access to the dedicated hardware?

Edit: Edited my question to indicate I'm referring specifically to the number of miners on the network.

I think Guillochon was thinking in terms of peers, not mining hardware.
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Nick ODell♦Apr 5 '13 at 17:25

He mentions "dedicated hardware". What else would he be referring to?
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BinaryMageApr 5 '13 at 17:29

Yes, I'm referring to the miners. Is the only benefit preventing takeover?
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GuillochonApr 5 '13 at 17:46

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The point of the distributed transaction verification system (the network of miners) is to prevent (or at least make as difficult as possible) takeover by any singular entity, yes. If there are any other benefits (relative to Bitcoin - obviously, electricity companies benefit), I'm not aware of them.
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BinaryMageApr 5 '13 at 17:53

If there were fewer miners, Bitcoin would be less reliable and you'd have to wait much longer to get the same level of security against a double-spending attack. We need as much mining as possible because mining secures the blockchain.

If I have 10 Bitcoins, I must be able to send them to Jeff. Otherwise, I don't have 10 Bitcoins. If I have 10 Bitcoins, I must be able to set them to Mary. Otherwise, I don't have 10 Bitcoins. But if I could send them to both Jeff and Mary, the system would break. So somehow, sending them to Jeff must prevent me from sending them to Mary too.

Bitcoin does this by having miners pick one of those transactions and heap a massive amount of computations on top of it. Among valid blockchains, Bitcoin prefers the one that likely took the most calculations to do. Miners pile calculations on top of the legitimate block chain to distinguish it from alternate chains. To "undo" a Bitcoin transaction, you must pile more computations on top of it than the rest of the miners combined piled on top of the legitimate chain that includes that transaction. The more mining, the faster you can accumulate sufficient calculations that undoing a transaction is impractical.

Miners are purchasing specialized hardware (known as ASICs) to mine for blocks more efficiently. This has no effect on the rate of block generation because the network auto adjusts difficulty every 2016 blocks (about 2 weeks).

By contrast, the blockchain can be hosted by any computer without specific hardware and is downloaded when you use a "full" client like bitcoin-qt. Because the blockchain is hosted everywhere, it would be near impossible to wipe it out, which is a good thing for a public ledger.

I guess that's more my question: Is there any positive to having as many miners as possible?
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GuillochonApr 5 '13 at 17:45

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There is an advantage to more miners when it comes to thwarting a 51% attack. Currently, the mining pool BTC Guild is approaching half the network hash rate, which would technically cede them control to fork the blockchain. It's unlikely that they'll do this, but some people are calling for miners to switch pools in an effort to prevent that threshold from being crossed.
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Taylor GerringApr 5 '13 at 18:01